MinT - Best Known (41−16, 41, s)-Nets in Base 27

Best Known (41−16, 41, s)-Nets in Base 27

(41−16, 41, 206)-Net over F₂₇ — Constructive and digital

Digital (25, 41, 206)-net over F₂₇, using

1 times m-reduction [i] based on digital (25, 42, 206)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (0, 2, 28)-net over F₂₇, using
    - kÂ =Â 2 construction [i]
  2. digital (0, 2, 28)-net over F₂₇ (see above)
  3. digital (0, 3, 28)-net over F₂₇, using
    - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
        the rational function field F₂₇(x) [i]
      - Niederreiter sequence [i]
  4. digital (0, 4, 28)-net over F₂₇, using
    - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
  5. digital (0, 5, 28)-net over F₂₇, using
    - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
  6. digital (0, 8, 28)-net over F₂₇, using
    - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
  7. digital (1, 18, 38)-net over F₂₇, using
    - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
        a function field by SÃ©mirat [i]

(41−16, 41, 266)-Net in Base 27 — Constructive

(25, 41, 266)-net in base 27, using

1 times m-reduction [i] based on (25, 42, 266)-net in base 27, using
- (u,Â u+v)-construction [i] based on
  1. (6, 14, 116)-net in base 27, using
    - 2 times m-reduction [i] based on (6, 16, 116)-net in base 27, using
      - base change [i] based on digital (2, 12, 116)-net over F₈₁, using
        net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
        a function field by SÃ©mirat [i]
  2. (11, 28, 150)-net in base 27, using
    - base change [i] based on digital (4, 21, 150)-net over F₈₁, using
      - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(41−16, 41, 2027)-Net over F₂₇ — Digital

Digital (25, 41, 2027)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(41−16, 41, 3136573)-Net in Base 27 — Upper bound on s

There is no (25, 41, 3136574)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 48519 328669 611716 453461 898562 392966 665769 389155 653851 794513 > 27⁴¹ [i]